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Nevanlinna Theory of Meromorphic Functions

  • Junjiro Noguchi
  • Jörg Winkelmann
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 350)

Abstract

The value distribution theory of meromorphic functions on C established by R. Nevanlinna in 1925 is described. It not only deepened complex function theory in one variable but also led to the covering theory by L. Ahlfors and the value distribution theory in several complex variables. The aim of this chapter is to introduce the most fundamental part of the theory in a self-contained manner. We try to give the proofs by making use of those by H.L. Selberg and H. Cartan so that they are straightforward and compatible with those in the case of several complex variables presented in the later chapters.

Keywords

Line Bundle Entire Function Meromorphic Function Zeta Function Theta Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

Ahlfors, L.V.

  1. [30]
    Beiträge zur Theorie der meromorphen Funktionen, 7e Congr. Math. Scand., Oslo, 1929, pp. 84–88, Oslo, 1930. Google Scholar
  2. [35]
    Zur Theorie der Überlagerungsflächen, Acta Math. 65 (1935), 157–194. MathSciNetCrossRefGoogle Scholar

Carlson, J. and Griffiths, P.

  1. [72]
    A defect relation for equidimensional holomorphic mappings between algebraic varieties, Ann. Math. 95 (1972), 557–584. MathSciNetCrossRefMATHGoogle Scholar

Cartan, H.

  1. [28]
    Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires, Ann. Sci. Éc. Norm. Super. 45 (1928), 255–346. MathSciNetMATHGoogle Scholar
  2. [29a]
    Sur la croissance des fonctions méromorphes d’une ou plusiers variables complexes, C. R. Math. Acad. Sci. Paris 188 (1929a), 1374–1376. MATHGoogle Scholar

Drasin, D.

  1. [77]
    The inverse problem of the Nevanlinna theory, Acta Math. 138 (1977), 83–151. MathSciNetCrossRefMATHGoogle Scholar

Griffiths, P. and King, J.

  1. [73]
    Nevanlinna theory and holomorphic mappings between algebraic varieties, Acta Math. 130 (1973), 145–220. MathSciNetCrossRefMATHGoogle Scholar

Hayman, W.K.

  1. [64]
    Meromorphic Functions, Oxford Math. Monog., Oxford University Press, London, 1964. MATHGoogle Scholar

Jensen, J.L.W.V.

  1. [1899]
    Sur un nouvel et important théorème de la théorie des fonctions, Acta Math. 22 (1899), 359–364. MathSciNetCrossRefMATHGoogle Scholar

Lang, S.

  1. [99]
    Complex Analysis, 4th ed., G.T.M. 103, Springer, Berlin, 1999. CrossRefMATHGoogle Scholar

Nevanlinna, F.

  1. [27]
    Über die Anwendung einer Klasse uniformisierender transzendenten zur Untersuchung der Wertverteilung analytischer Funktionen, Acta Math. 50 (1927), 159–188. MathSciNetCrossRefMATHGoogle Scholar

Nevanlinna, R.

  1. [25]
    Zur Theorie der meromorphen Funktionen, Acta Math. 46 (1925), 1–99. MathSciNetCrossRefMATHGoogle Scholar
  2. [29]
    Le Théorème de Picard-Borel et la Théorie des Fonctions Méromorphes, Gauthier-Villars, Paris, 1929. MATHGoogle Scholar
  3. [53]
    Eindeutige Analytische Funktionen, Grundl. Math. Wiss. 46, Springer, Berlin, 1953. CrossRefMATHGoogle Scholar

Ozawa, M.

  1. [76]
    Modern Function Theory I (in Japanese), Morikita Publ. Co., Tokyo, 1976. Google Scholar

Shimizu, T.

  1. [29]
    On the theory of meromorphic functions, Jpn. J. Math. 6 (1929), 119–171. MATHGoogle Scholar

Weitsman, A.

  1. [72]
    A theorem of Nevanlinna deficiencies, Acta Math. 128 (1972), 41–52. MathSciNetCrossRefMATHGoogle Scholar

Weyl, H. and Weyl, J.

  1. [38]
    Meromorphic curves, Ann. Math. 39 (1938), 516–538. MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  • Junjiro Noguchi
    • 1
    • 2
  • Jörg Winkelmann
    • 3
  1. 1.The University of TokyoTokyoJapan
  2. 2.Tokyo Institute of TechnologyTokyoJapan
  3. 3.Ruhr-University BochumBochumGermany

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